Lecture 22. December 19, Department of Biostatistics Johns Hopkins Bloomberg School of Public Health Johns Hopkins University.

Transcription

1 Lecture 22 Department of Biostatistics Johns Hopkins Bloomberg School of Public Health Johns Hopkins University December 19, 2007

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3 1 tests for equivalence of two binomial 2 tests for, 2 2 tables 3 tests for multiple binomial 4 tests for, r c tables 5 tests for goodness of fit

4 An alternative approach to uses the chi-squared statistic (Observed Expected) 2 Expected Observed are the observed counts Expected are the expected counts under the null hypothesis The sum is over all four cells This statistic follows a distribution with 1 df The statistic is exactly the square of the difference in Score statistic

5 Example Trt Side Effects None Total X Y p 1 and p 2 are the cure rates H 0 : p 1 = p 2

6 The χ 2 statistic is (O E) 2 E O 11 = 44, E 11 = O 21 = 77, E 21 = O 12 = 56, E 12 = O 22 = 43, E 22 = χ 2 = (44 55) = = = = 54 (77 66) (56 45) (43 54)2 54 Which turns out to be Compare to a χ 2 with one degree of freedom (reject for large values). pchisq(8.96, 1, lower.tail = FALSE) #result is 0.002

7 R code dat <- matrix(c(44, 77, 56, 43), 2) chisq.test(dat) chisq.test(dat, correct = FALSE)

8 Notation reminder n 11 = x n 12 = n 1 x n 1 = n 1+ n 21 = y n 22 = n 2 y n 2 = n 2+ n +1 n +2

9 Notes Reject if the statistic is too large Alternative is two sided Do not divide α by 2 A small χ 2 statistic implies little difference between the observed values and those expected under H 0 The χ 2 statistic and approach generalizes to other kinds of tests and larger contingency tables Alternative computational form for the χ 2 statistic χ 2 = n(n 11n 22 n 12 n 21 ) 2 n +1 n +2 n 1+ n 2+

10 Notice that the statistic: χ 2 = n(n 11n 22 n 12 n 21 ) 2 n +1 n +2 n 1+ n 2+ Notes does not change if you transpose the rows and the columns of the table Surprisingly, the χ 2 statistic can be used the rows are fixed (binomial) the colums are fixed (binomial) the total sample size is fixed (multinomial) none are fixed (Poisson) For a given set of data, any of these assumptions results in the same value for the statistic

11 Maternal age versus birthweight 1 Cross-sectional sample, only the total sample size is fixed Birthweight Mat. Age < 2500g 2, 500g Total < 20y y Total H 0 : MA is independent of BW H a : MA is not independent of BW 1 From Agresti Categorical Data Analysis second edition

12 Continued Under H 0 (est) P (MA < 20) = =.25 Under H 0 (est) P (BW < 2500) = =.125 Under H 0 (est) P (MA < 20 and BW < 2, 500) = Therefore E 11 = E 12 = E 21 = E 22 = = = = = χ 2 = ( ) ( ) Compare to critical value qchisq(.95, 1)=3.84 ( ) ( ) = 6.86 Or calculate P-value pchisq(6.86, 1, lower.tail = F)=.009

13 cont d 2 Alcohol use Group High Low Total Clergy Educators Executives Retailers Total ,200 2 From Agresti s Categorical Data Analysis second edition

14 Interest lies in whether or not the proportion of high alcohol use is the same in the four occupations H 0 : p 1 = p 2 = p 3 = p 4 = p H a : at least two of the p j are unequal O 11 = 32, E 11 = O 12 = 268, E 12 = statistic (0 E) 2 E = df = (Rows 1)(Columns 1) = 3 Pvalue pchisq(20.59, 3, lower.tail = FALSE) 0

15 Word distributions Book Word Total a an this that with without Total From Rice Mathematical Statistics and Data Analysis, second edition

16 H 0 : The probabilities of each word are the same for every book H a : At least two are different O 11 = 147 E 11 = O 12 = 186 E 12 = (O E) 2 E = df = (6 1)(3 1) = 10

17 Wife s Rating Husband N F V A Tot N F V A N=never, F=fairly often, V=very often, A=almost always 4 4 From Agresti s Categorical Data Analysis second edition

18 cont d H 0 : H and W ratings are independent H a : not independent P(H = N & W = A) = P(H = N)P(W = A) stat = (O E) 2 E O 11 = 7 E 11 = = 2.51 E ij = n i+ n +j /n df = (Rows 1)(Cols 1)

19 cont d x<-matrix(c(7,7,2,3, 2,8,3,7, 1,5,4,9, 2,8,9,14),4) chisq.test(x) (O E) 2 E = df = (4 1)(4 1) = 9 p value =.049 Cell counts might be too small to use large sample approximation

20 Notes Equal distribution and test yield the same results Same test results if row totals are fixed column totals are fixed total ss is fixed none are fixed Note that this is common in statistics; mathematically equivalent results are applied in different settings, but result in different interpretations

21 result requires large cell counts df is always (Rows 1)(Columns 1) s of Fishers exact test can be used or continuity corrections can be employed

22 Exact permutation test Reconstruct the individual data W:NNNNNNNFFFFFFFVVAAANNFFFFFFFF... H:NNNNNNNNNNNNNNNNNNNFFFFFFFFFF... Permute either the W or H row Recalculate the contingency table Calculate the χ 2 statistic for each permutation Percentage of times it is larger than the observed value is an exact P-value chisq.test(x, simulate.p.value = TRUE)

23 goodness of fit Results from R s RNG [0,.25) [.25,.5) [.5,.75) [.75, 1) Total Count TP H 0 : p 1 =.25, p 2 =.25, p 3 =.25, p 4 =.25 H a : any p i it s hypothesized value

24 Continued O 1 = 254 E 1 = = 250 O 2 = 235 E 2 = = 250 O 3 = 267 E 3 = = 250 O 4 = 244 E 4 = = 250 (O E) 2 E = df = 3 P value =.52

25 Mendel s hypothesis Phenotype Yellow Green Total Observed TP Expected H 0 : p 1 =.75, p 2 =.25 (0 E) 2 E = ( ) ( ) =.015

26 Continued df = 1 P-value =.90 Fisher combined of Mendel s tables χ 2 v i χ 2 vi Statistic 42, df = 84, P-value = Agreement with theoretical counts is perhaps too good?

27 Notes on GOF Test of whether or not observed counts equal theoretical values Test statistic is (0 E) 2 E TS follows χ 2 distribution for large n df is the number of cells minus 1 Undirected alternative is problematic Especially useful for RNGs Kolmogorov/Smirnov test is an alternative test that does not require discretization but often has low power